Chapter 7
The Time
Value of
Money
1

Time Value
A. Process of expressing
1. The present value of $1 invested now in future terms. (Compounding)
Compounding – Process by which interest is paid on interest that was previously earned.
2. The future value of $1 invested in terms of the present
Future Value of a dollar – Amount to which a single payment will grow at some rate of interest.
B. Payments are either:
1. a single payment
2. A series of equal payments (an annuity)
2

Time Value
C. Time value of money problems may be solved by using:
1. Interest tables
2. Financial calculators
3. Software
3

Variables for Time Value of
Money Problems
 PV = present value
 FV = future value
 PMT = annual payment
 N = number of time periods
 I = interest rate per period
4

Future Value
 Future value of $1 takes a single payment in the present into the future.
 General equation for the future value of $1:
P
0
(1 + i
) n = P n
FV = PV (1 + i) n
5

Future Value Illustrated
 PV = -100
 I = 5
 N = 20
 PMT = 0
 FV = ?
 = 265.30
6

Greater Terminal Values
 Higher interest rates
 Longer time periods
 Result in greater terminal values
7

Greater Terminal Values
8

Present Value
 Present value of $1 brings a single payment in the future back to the present.
Present Value – Current value of a dollar to be received in the future.
Discounting – Process of determining present value.
9

Present Value
 General equation for the present value of $1:
P
0
= P n
(1+ i
) n
PV = FV [ 1 / (1 + i) n ]
1.
PV = Present Value
2.
FV = Future Value
3.
i = Interest Rate per Period
4.
n = Number of Compounding Periods
10

Present Value Illustrated
 FV = 100
 I = 6
 N = 5
 PMT = 0
 PV = ?
 = -74.73
11

Lower Present Values
 Higher interest rates
 Longer time periods
 Result in lower present values
12

Lower Present Values
13

Financial Calculators and Excel
 Express the cash inputs (PV, FV, and PMT) as cash inflows and cash outflows
 At least one of the cash variables must be
–an inflow (+)
–an outflow (-)
14

Simple Interest
 Simple Interest – No Compounding
– Good to have if you withdraw interest each period.
– SI = Principal (PV) x rate (i) x time (n)
15

Future Value of a Single
Amount Example
 You buy a stock for $10 and expect the price to increase 9 percent annually. After 10 years, what is the anticipated price of the stock?
16

Future Value of a Single Amount
Example
 The unknown: FV
 The givens:
–PV = 10
–PMT = 0
–N = 10
–I = 9
The answer:
$23.67
17

FV Interpretation
 A $10 stock will be worth $23.67 after 10 years if its price grows 9% annually.
18

Present Value of a Single
Amount Example
 What is the cost of a stock that was sold for $23.67, held for 10 years and whose value appreciated 9 percent annually?
19

Present Value of a Single
Amount Example
 The unknown: PV
 The givens:
–FV = 23.67
–PMT = 0
–N = 10
–I = 9
The answer:
$10
20

Interpretation
 $23.67 received after ten years is worth $10 today if the rate of return is 9 percent.
21

Interpretation of Future and
Present Values
These two problems are mirror images:
 In the first case, the $10 is compounded into its future value ($23.67).
 In the second case, the future value
($23.67) is discounted back to its present value ($10).
22

Rate of Return Example
 A stock was purchased for $10 and sold for $23.67 after 10 years. What was the return?
23

Future Determination of the Interest Rate
(can use Present too)
 The unknown: I
 The givens:
–PV = 10
–PMT = 0
–N = 0
–FV = 23.67
The answer:
9%
24

Interpretation
 The yield on a $10 investment that was sold after 10 years for $23.67 is 9%.
25

Non-annual Compounding
 More than one interest payment a year
 State Interest rates are always annual interest rates.
 More frequent compounding
26

Non-annual Compounding
 Multiply number of years by frequency of compounding
 Divide interest rate by frequency of compounding
27

Periods less than One Year
 Same variables as in all time value problems except N < 1.
 Calculate by dividing the number of days by 365.
28

Illustration for Return on
Investment
 What is the return on an investment that costs $98,543 and pays
$100,000 after 45 days?
29

Determination of Return
 The unknown: I
 The givens:
–PV = -98,543
The answer:
12.64%
–N = 0.1233 (45/365)
–FV = 100,000
–PMT = 0
30

Interpretation
 $98,543 invested for 45 days grows to $100,000 at 12.64 percent.
31